April 6-10, 2009 Week #10 Practice Quiz Name:
Math 412
Problem 1. (5 points each) Find all values of zthat satisfy the following equations:
(a): sin z= 2
[Hint: arcsin(z) = −ilog iz + (1 −z2)1/2.]
(b): z= (−1)1/π
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Math 412 Week #10 Practice Quiz: Solutions for Problems 1, 2, and 3 - Prof. Scott Annin, Quizzes of Mathematics
Solutions for problems 1, 2, and 3 from the math 412 week #10 practice quiz. Problem 1 asks to find all values of z that satisfy sin z = 2, using the given hint. Problem 2 requires proving that f'(z) = αz · log(α) for the principal value of αz. Lastly, problem 3 asks to prove the identity of trigonometric functions: sin z1 − sin z2 = 2 cos ((z1 + z2) / 2) · sin ((z1 − z2) / 2).
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April 6-10, 2009 Week #10 Practice Quiz Name:
Math 412
Problem 1. (5 points each) Find all values of z that satisfy the following equations:
(a): sin z = 2
[Hint: arcsin(z) = −i log
[
iz + (1 − z 2 ) 1 / 2
]
.]
(b): z = (−1)^1 /π
Problem 2. (4 points) Let α ∈ C ∗ be fixed. Let f (z) = α z denote the principal value of α z .
Prove that
f ′ (z) = α z · Log(α).
Problem 3. (6 points) Prove that for all z 1 , z 2 ∈ C,
sin z 1 − sin z 2 = 2 cos
z 1 + z 2
2
sin
z 1 − z 2
2